Peter Langer

The task of solving the stationary Schrödinger equation is a longstanding and enormous challenge in many important areas of natural sciences. As explicit symbolic solutions of the operator eigenvalue problem are only attainable in very rare cases, one mostly has to resort to numerical techniques, especially to methods for giant Hermitian eigenvalue problems. In this thesis we are concerned with the specific case of triatomic molecules that exhibit the so called Double-Renner effect. To begin with, we explain the origin and the theoretical background of the abstract Schrödinger problem and we discuss viable techniques for the transition to suitable finite dimensional Hermitian matrices that approximate the original Hamiltonian in a reasonable fashion, and thus, make it accessible for a numerical treatment. However, due to tremendous storage requirements and computing times, that may soon extend to a couple of weeks, the use of conventional so-called direct solvers (QR method, RRR algorithm) is either not feasible or not advisable. Therefore, our main focus for the treatment of the matrix eigenvalue problem is on Jacobi-Davidson type methods that belong to the alternative class of iterative projection algorithms. Our aim is to show that these methods may be successfully applied in our context, in the sense that they are more efficient in terms of computing time than direct eigensolvers on the one hand, and the fellow algorithms of the iterative projection method class (Lanczos, Davidson, Olsen) on the other hand. To do so, we have to construct and to identify suitable preconditioners for the arising shift-and-invert systems that take advantage of the inherent information of the specific problem. Besides, efficient and problem-adjusted routines for matrix-vector multiplication are decisive for the success of our approach. Our ideas are illustrated and confirmed by extensive numerical experiments and results.